Approximation of the scattering cross section for aggregated spherical particles

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Approximation of the scattering cross section for

aggregated spherical particles

Frédéric Gruy, Sandra Jacquier

To cite this version:

Frédéric Gruy, Sandra Jacquier. Approximation of the scattering cross section for aggregated spherical particles. 9th International Conference on Electromagnetic and Light Scattering by Non-Spherical Particles: Theory, Measurements, and Applications„ May 2006, Saint-Petersbourgh, Germany. pp.51-54. �hal-00132156�

Approximation of the scattering cross section for aggregated spherical particles

Gruy Frédéric(1)_{, Jacquier Sandra}(1)

(1) _{Ecole Nationale Supérieure des Mines de Saint-Etienne Centre SPIN (LPMG, UMR}

CNRS 5148)

Département GENERIC, 158 Cours Fauriel 42023 Saint-Etienne Cedex 02 e-mail: gruy@emse.fr, jacquier@emse.fr

ABSTRACT

Exact solution for scattering cross section coming from Xu’s computations is compared with different approximations.

1INTRODUCTION

Many industrial operations involve solid-liquid suspensions. During the corresponding processes suspensions are put in motion. As a consequence, collisions and sticking of particles, i.e aggregation of particles, occurs. This phenomenon is due to Brownian motion in the case of sub-micronic particles and due to the velocity field heterogeneity in the case of larger particles. The aggregates formed usually contain a few primary particles because large aggregates undergo fragmentation leading to a limit size for the aggregates. So, primary spherical particles of metallic oxides (ZrO2, TiO2, Al2O3, SiO2)

are used as precursors in ceramic industry. The corresponding suspensions contain aggregates, each one composed of less than one hundred primary particles. The on-line characterization of the suspension is made by spectral turbidimetry [1], i.e. attenuation of a light beam by particle scattering and absorption. In order to analyze the optical signal, i.e. turbidity, the scattering cross sections of aggregates are needed. An important literature is available about the calculation of the light scattering cross section of aggregates. For instance, Xu and Khlebtsov [2] (for more information see the references therein) proposed an exact calculation based on a generalization of the Mie theory. So, the scattering cross sections depend on the size of the primary particles, on the morphology of the aggregates and on the refractive index of the material and of the medium. This exact, but complicated theory cannot be easily used to interpret, for instance, turbidity spectra on-line. This is due to the time consuming calculations for a large amount of different aggregates (doublet, triplet...) which are produced during the aggregation process. The topic of this paper is to propose approximations (averaged over all the orientations) for scattering cross section corresponding to aggregates of few spherical and non absorbing particles. As the accuracy of turbidity measurements is more than 5%, we will consider an approximation as useful and valid if it describes the reality within this error.

2SEVERAL APPROXIMATIONS FOR SCATTERING CROSS SECTIONS

An aggregate is composed of N primary spherical particles. The radius and the dimensionless size parameter of the primary particle are respectively denoted a1 and

1 1 2 a λ π

α = . Two kinds of modeling can be used to calculate approximately the optical

properties of porous materials:

Effective refractive index method. An aggregate is characterized by its effective

diameter aN;e and its mean inner volume fraction a Na a3N,e

3 1 )/ ( =

determine the optical properties of an aggregate is to calculate its effective refractive

index ma (see, for instance, [3]). The equation derived by Maxwell-Garnett has been

proved to be suitable in certain case

2 1 2 1 2 2 2 2 + − Φ = + − m m m m a a a _{, (1)}

where m and ma are the relative refractive index for primary particles and aggregates,

respectively. With known diameter and effective refractive index of aggregate, the Mie theory [4] allows us to calculate the scattering cross section CN for a given wavelength. Two particular cases can be considered for choosing the effective diameter:

9 the compact sphere: the non porous sphere with the same mass of material as in the

aggregate. Thus,

a N;e= N1/3 a1 (Φa=1and ma = m); (2)

9 the “projected area” equivalent sphere: in aggregation theory and optical properties modeling, the relevant parameter is the projected area Sp of the object on a plane.

Thus, we define the radius aN;e of the equivalent sphere for an aggregate as:

0 , 2 p S a Ne = π , (3) where 0 p

S is the average projected area according to all aggregate orientations.

Soft-particle methods. Generally, the object (primary particle, aggregate ...) can be

divided into smaller identical parts (elements). Each element is polarisable. In the presence of a variable electric field, the element becomes an oscillating dipole, which itself creates an electromagnetic field. When an object is illuminated by an electromagnetic wave, each element receives the incident electric field and the one coming from the other elements. As a result, one may associate an oscillating dipole moment to each element. Thus, the object emits an electromagnetic wave (scattered wave), which includes the contribution of each oscillating dipole.

Most often, the incident wave is randomly polarized and the object (scatterer) can randomly orientate. Thus, the optical properties are obtained after calculating an average over all the wave polarization states and object orientations.

Several models were published, each one characterized by the polarisable element, the object, the calculation type. For instance, Berry and Percival [5], Khlebtsov [6] and Xu and Khlebtsov [2] calculate the aggregate optical properties, if the polarisable element are respectively: Rayleigh scatterer, Rayleigh-Debye-Gans (RDG) scatterer and Mie scatterer.

So, the Khlebtsov’s [6] procedure, similar to Berry-Percival one, has been applied to aggregates of many particles (fractal aggregates). RDG scatterers are characterized by

1 −

m << 1 and 4 a₁m−1 λ

π

<< 1. Then, the scattering cross section obeys:

∫

∫

where Ω, ϑ , q are the solid angle, the scattering angle and the scattering vector,

) (

1 ϑ

F is proportional to the phase function corresponding to the primary particle

(sphere). The structure factor S(q) depends on the aggregate morphology:

S(q) = 2. (5) , 1 / ) /( sin(qR qR N N N j i j i ij ij ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +

∑

Here, Ri;j is the distance between the centers of the primary particules i and j inside the

aggregate. However, when 4 a₁m−1

λ π

is not small, the approximation of anomalous diffraction [4] can be applied. In the Rayleigh-Debye-Gans domain, there is interference of electromagnetic waves which are independently scattered by all small volume elements. In the anomalous di_raction domain, there is straight transmission and subsequent diffraction. In this case, the scattered intensity is concentrated near the original direction of propagation and the scattering cross section obeys the relation [4]:

∫

∫

propagation direction. δ is the path of light through the object. This calculated path is a function of the projection coordinates. This kind of calculation was previously achieved for fractal clusters by Khlebtsov [7].

3NUMERICAL INVESTIGATION

We studied aggregates with number of particles N = 2; 4; 8; 16; 64; 100. The size parameter of the primary particle is in the range [0.013 ; 9.25]. For each N-aggregate, we consider several morphologies or compactness: chain, plane, cube... Tested materials are the ones with low (SiO2, m = 1.08), intermediate (Al2O3, m = 1.32) and high (TiO2, m =

1.94) optical contrast in water. We compare exact values of scattering cross section derived from Xu’s theory [2] with the corresponding one from the above-described approximations: compact sphere (CS) (Equations. (1), (2)); effective refractive index (ERI) method (Equations. (1), (3)); Percival-Berry-Khlebtsov (PBK) method (Equations. (4), (5)); anomalous diffraction (AD) approximation (Equation (6)). Results are shown as the ratio of actual scattering cross section to the sum of aggregate primary particles cross sections CN/(NC1).

a b

Consider main features of mean scattering cross sections of aggregates like it was done in [8]. Figures 1(a,b) represent RXu = CXu,N/(NCMie,;1) against the primary particle size

parameter for several N-chains and for the two systems TiO2/water and SiO2/water. The

trends are the same for compact aggregates (the compact configuration means for example a tetrahedron for the case of 4 primary particles, a cube for the case of 8 and so on). The shape of the curves RXu is similar whatever the material. RXu is a decreasing

function of the size parameter. For very small size parameter RXu tends to N as expected

by [4]. For high value of the size parameter RXu tends to a value in the range [0-1]. When

we consider the data for all the aggregates we may divide the α-range into two sub-ranges: [0,2] with sharp decrease of RXu and [2,10] with a nearly constant value of RXu.

In order to compare the different approximate methods we consider for each method ratio Rmethod / RXu with Rmethod = Cmethod,N/(NCmethod,1). Tables 1 and 2 contain the

performances of each method as the mean value of R (over 100 α-values) and the mean

relative deviation (σ) between Rmethod and RXu. With the view to obtain significant

calculations over different size parameter, we restricted the number of the primary particles by the values of 2, 4, 8 and 16. We also performed calculations in the case of 64 and 100 primary particles but with less number of size parameter (CPU time is too long) that why there are not include in the values of these tables.

Table 1: 0 < α < 2

configuration linear compact

index SiO2 AlO2O3 TiO2 SiO2 AlO2O3 TiO2

method mean σ mean σ mean σ mean σ mean σ mean σ

CS 1.73 0.95 1.72 0.95 1.39 0.88 1.20 0.24 1.21 0.25 1.07 0.32 ERI 1.23 0.35 1.16 0.31 1.00 0.18 1.02 0.05 0.99 0.05 0.97 0.12 PBK 0.99 0.02 0.96 0.06 0.811 _0.211 _{0.99 0.01 0.98 0.07 0.85}1 _0.171 AD 0.60 0.45 0.57 0.48 0.57 0.53 0.56 0.50 0.53 0.52 0.51 0.57 RXu 2.47 2.32 2.56 2.45 2.85 3.08 3.70 4.12 3.82 4.34 3.97 5.06 (1) _{0 < α < 1} Table 2: 0 < α <10

configuration linear compact

index SiO2 AlO2O3 TiO2 SiO2 AlO2O3 TiO2

method mean σ mean σ mean σ mean σ mean σ mean σ

CS 1.36 0.45 0.69 0.44 0.64 0.43 1.14 0.16 0.78 0.29 0.99 0.68 ERI 0.95 0.09 0.97 0.09 0.98 0.10 0.96 0.05 0.98 0.16 1.25 0.91 PBK 1.08 0.13 1.51 0.62 1.462 _0.562 _{1.12 0.22 2.57 2.19 3.07}2 _4.262 AD 0.94 0.10 1.07 0.16 1.07 0.08 0.94 0.07 1.05 0.12 1.44 1.27 RXu 1.19 0.25 0.87 0.18 0.83 0.18 1.40 0.50 0.76 0.38 0.64 0.43 (2) _{1 < α < 10} 4CONCLUSION

As expected CS method is not appropriate. AD (anomalous diffraction) is a good approximation for 2 < α < 10. PBK is a good approximation for 0 < α < 2 but less efficient for material with high optical contrast. ERI seems the best for 0 < α < 10 except for chains of small primary particles. The last row of the Table 1 and 2 shows RXu. As it is

expected, this approximation is not acceptable for small primary particles (0 < α < 2). The result becomes better for large primary particles, especially in the case of chains, but anyway ERI remains the best approximation.

ACKNOWLEDGMENTS

We would like to thanks Dr. Y.-l. Xu for making his GMM code available as shareware.

REFERENCES

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